Volume of a Cone
The volume of a cone.
Formula
Variables
Example
Radius 3, height 6: V = (1/3)π·3²·6 ≈ 56.5 cubic units.
How it works
A cone’s volume is one-third of the volume of a cylinder with the same base radius and height. That relationship comes from the way a cone narrows smoothly to a point, so it occupies less space than a full cylinder even though both share the same base area and height. The formula first finds the base area, πr², then multiplies by height h, and finally takes one-third of that amount.
When to use it
Use this formula whenever you need the amount of space inside a cone-shaped object, such as an ice cream cone, funnel, or conical tank. It also appears in geometry problems and engineering contexts involving cones, pyramids, or tapered shapes.
Worked example, step by step
Given radius r = 4 and height h = 9 Step 1: Substitute into the formula V = (1/3)πr²h Step 2: V = (1/3)π(4²)(9) Step 3: V = (1/3)π(16)(9) Step 4: V = 48π Step 5: V ≈ 150.8 Final answer: 150.8 cubic units
Common mistakes to avoid
- Using the diameter instead of the radius in the formula
- Forgetting to square the radius before multiplying by height
- Leaving out the factor of one-third
Did You Know?
A cone holds exactly one-third the volume of the cylinder that encloses it — pour water to see it takes three cones to fill.
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